Prove-that-a-2-b-2-c-2-ab-bc-ca-a-b-c-R-

Question Number 32768 by Rasheed.Sindhi last updated on 01/Apr/18

Prove that a^{2} + b^{2} + c^{2} ⩾ a b + b c + c a

\forall a, b, c \in R

Answered by mrW2 last updated on 02/Apr/18

a^{2} + b^{2} - 2 a b = {(a - b)}^{2} ⩾ 0

a^{2} + b^{2} ⩾ 2 a b

b^{2} + c^{2} ⩾ 2 b c

c^{2} + a^{2} ⩾ 2 c a

Σ :

2 (a^{2} + b^{2} + c^{2}) ⩾ 2 (a b + b c + c a)

a^{2} + b^{2} + c^{2} ⩾ a b + b c + c a

Commented by Joel578 last updated on 02/Apr/18

In 3 rd line, did you mean

c^{2} + a^{2} ⩾ 2 c a ?

Commented by Rasheed.Sindhi last updated on 02/Apr/18

Th \forall n k s Sir!

Commented by mrW2 last updated on 02/Apr/18

y e s . t h a n k s!

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